Some invariant connections on symplectic reductive homogeneous spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911028098367488 |
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| author | Abouqateb, Abdelhak Dani, Othmane |
| author_facet | Abouqateb, Abdelhak Dani, Othmane |
| contents | A symplectic reductive homogeneous space is a pair $(G/H,Ω)$, where $G/H$ is a reductive homogeneous $G$-space and $Ω$ is a $G$-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric spaces, and flag manifolds. This paper focuses on the existence of a natural symplectic connection on $(G/H,Ω)$. First, we introduce a family $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$ of $G$-invariant connection on $G/H$, and establish that $\nabla^{0,1}$ is flat if and only if $(G/H,Ω)$ is locally a symplectic Lie group. Next, we show that among all $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$, there exists a unique symplectic connection, denoted by $\nabla^\mathbf{s}$, corresponding to $a=b=\tfrac{1}{3}$, a fact that seems to have previously gone unnoticed. We then compute its curvature and Ricci curvature tensors. Finally, we demonstrate that the $\operatorname{SU}(3)$-invariant preferred symplectic connection of the Wallach flag manifold $\operatorname{SU}(3)/\mathbb{T}^2$ (from Cahen-Gutt-Rawnsley) coincides with the natural symplectic connection $\nabla^\mathbf{s}$, which is furthermore Ricci-parallel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some invariant connections on symplectic reductive homogeneous spaces Abouqateb, Abdelhak Dani, Othmane Differential Geometry Mathematical Physics Symplectic Geometry 53D05, 53B05, 14M17, 53C30, 70G45 A symplectic reductive homogeneous space is a pair $(G/H,Ω)$, where $G/H$ is a reductive homogeneous $G$-space and $Ω$ is a $G$-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric spaces, and flag manifolds. This paper focuses on the existence of a natural symplectic connection on $(G/H,Ω)$. First, we introduce a family $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$ of $G$-invariant connection on $G/H$, and establish that $\nabla^{0,1}$ is flat if and only if $(G/H,Ω)$ is locally a symplectic Lie group. Next, we show that among all $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$, there exists a unique symplectic connection, denoted by $\nabla^\mathbf{s}$, corresponding to $a=b=\tfrac{1}{3}$, a fact that seems to have previously gone unnoticed. We then compute its curvature and Ricci curvature tensors. Finally, we demonstrate that the $\operatorname{SU}(3)$-invariant preferred symplectic connection of the Wallach flag manifold $\operatorname{SU}(3)/\mathbb{T}^2$ (from Cahen-Gutt-Rawnsley) coincides with the natural symplectic connection $\nabla^\mathbf{s}$, which is furthermore Ricci-parallel. |
| title | Some invariant connections on symplectic reductive homogeneous spaces |
| topic | Differential Geometry Mathematical Physics Symplectic Geometry 53D05, 53B05, 14M17, 53C30, 70G45 |
| url | https://arxiv.org/abs/2506.23211 |